Multiplicative updates in coordination games and the theory of evolution

In this paper we point out a new and unexpected connection between three fields: Evolution Theory, Game Theory, and Algorithms. In particular, we study the standard equations of population genetics for Evolution, in the presence of recombination (sex), focusing on the important special case of weak selection [1,2] in which all fitness values are assumed to be close to one another. Weak selection is the mathematical regime capturing the widely accepted Neutral Theory proposed by Kimura in the 1970s [3], hypothesizing that evolution proceeds for the most part not by substantial increases in fitness but by essentially random drift. We show that in this regime evolution through natural selection and sex is tantamount to a game played through the multiplicative weight updates game dynamics [4]. The players of the game are the genes (genetic loci), the strategies available to each player are the alleles of the gene, and the probabilities whereby a player plays a strategy is the strategy's frequency in the population. The utility to each player/gene of each strategy profile is the fitness of the corresponding genotype (organism). That is, the game is a coordination game between genes, in which the players' interests are perfectly aligned. Importantly, the utility maximized in this game, as well as the amount by which each allele is boosted, is precisely the allele's mixability, or average fitness, a quantity recently proposed in [5] as a novel concept that is crucial in understanding natural selection under sex, thus providing a rigorous demonstration of that insight. We also establish a result regarding the maintenance of genetic diversity (multiplicity of alleles per gene). We prove that the equilibria in two-person coordination games are likely to have large supports, and thus genetic diversity need not suffer much at equilibrium. Establishing large supports involves answering through a novel technique the following question: what is the probability that for a random square matrix $A$ (with entries drawn independently from smooth distributions that are symmetric around zero) both systems Ax=1 and ATy=1 have positive solutions? The proof is through a simple potential function argument. Both the question and the technique may be of broader interest. It has often seemed astonishing --- even to experienced students of Evolution, Darwin included --- that the crude mechanism of natural selection is responsible for producing the dazzling variety of Life around us. The present mathematical connection of Evolution with the multiplicative weight updates algorithm --- a technique that has surprised our field time and again with its fantastic effectiveness and versatility --- may carry some explanatory force in this regard.